COST-EFFICIENT SCHEDULING FOR DEADLINE CONSTRAINED …sina.sharif.ac.ir › ~movaghar ›...

Computing and Informatics, Vol. 37, 2018, 838–864, doi: 10.4149/cai 2018 4 838

COST-EFFICIENT SCHEDULING FOR DEADLINECONSTRAINED GRID WORKFLOWS

Alireza Dehlaghi-Ghadim

School of Electrical and Computer EngineeringUniversity of TehranTehran, Irane-mail: [email protected]

Reza Entezari-Maleki

School of Computer ScienceInstitute for Research in Fundamental Sciences (IPM)Tehran, Irane-mail: [email protected]

Ali Movaghar

Department of Computer EngineeringSharif University of TechnologyTehran, Irane-mail: [email protected]

Abstract. Cost optimization for workflow scheduling while meeting deadline isone of the fundamental problems in utility computing. In this paper, a two-phasecost-efficient scheduling algorithm called critical chain is presented. The proposedalgorithm uses the concept of slack time in both phases. The first phase is deadlinedistribution over all tasks existing in the workflow which is done considering criti-cal path properties of workflow graphs. Critical chain uses slack time to iterativelyselect most critical sequence of tasks and then assigns sub-deadlines to those tasks.In the second phase named mapping step, it tries to allocate a server to each taskconsidering task’s sub-deadline. In the mapping step, slack time priority in select-

Cost-Efficient Scheduling for Deadline Constrained Grid Workflows 839

ing ready task is used to reduce deadline violation. Furthermore, the algorithmtries to locally optimize the computation and communication costs of sequentialtasks exploiting dynamic programming. After proposing the scheduling algorithm,three measures for the superiority of a scheduling algorithm are introduced, andthe proposed algorithm is compared with other existing algorithms considering themeasures. Results obtained from simulating various systems show that the proposedalgorithm outperforms four well-known existing workflow scheduling algorithms.

Keywords: Grid computing, workflow, slack time, critical path, cost-based schedul-ing

Notations

T Set of all tasks in the applicationE Set of all dependencies in the applicationti Task ieij Dependency between task i and task jδ Deadline of the applicationS Set of serverssi Server iTij Processing time of task ti on server sjCij Processing cost of task ti on server sjS(ti) The server allocated to execute task tiTiS(ti) Execution time of task ti on S(ti)Delay(ti, tj) Data transmission time between task ti and task tj

on the link between S(ti) and S(tj)imm preds(ti) All tasks in the workflow graph in which ti is their

immediate successorimm succs(ti) All tasks in the workflow graph in which ti is their

immediate predecessorMET(ti) Minimum time for the execution of task ti on the fastest

serverMTT(eij) Minimum data transmission time between task ti

and task tjEST(ti) Earliest start time of task tiLFT(ti) Latest finish time of task tiST(ti) Slack time of task tiCST Chain start timeCFT Chain finish timeSST Schedule start timeSFT Schedule finish timeNC Normalized costθ Deadline factorTmin Minimum execution time of the application

840 A. Dehlaghi-Ghadim, R. Entezari-Maleki, A. Movaghar

1 INTRODUCTION

Grid is an infrastructure with the aim of solving high scale problems in science,economy, aerology, engineering and many other fields [1]. Resource sharing is one ofthe most significant advantages of grid computing [2]. The most important resourcesshared in grids include CPU, main memory, secondary memory, network bandwidth,and data. Traditional resource management techniques provide no incentive forusers to share resources fairly. Consequently, to support different levels of Quality ofService (QoS) and manage priority between user applications, utility grid computinghas been emerged [3, 4]. In this paradigm, users have to pay for each time they useservers with specific QoS. How to allocate grid servers to the tasks to satisfy thespecific needs is one of the important challenges in this area.

This paper focuses on workflow scheduling with the aid of heuristics. In this case,workflows are composed of several tasks with partial order, in the way that sometasks have control or data dependencies on the others. Many complex applications indifferent domains such as e-science as bioinformatics and astronomy, and e-businesscan be modeled as workflows [5]. To solve the applications, the resulted workflowsneed to be processed, so the tasks should be executed based on their dependen-cies [6]. We can describe workflows with Directed Acyclic Graphs (DAGs) in whicheach node in DAG represents a specific task in the corresponding workflow. There-fore, the scheduling problem can be stated as assigning a DAG of tasks to the limitedprocessing units according to their requirements and transposition constraints. Tosolve this type of scheduling problems, two different approaches can be used: ap-proximation and heuristic. In the approximate algorithms, since it is unlikely thatthere can ever be efficient polynomial-time exact algorithms solving NP-hard prob-lems, one settles for polynomial-time sub-optimal solutions so called approximation,which uses formal computational models to obtain sufficiently good solution insteadof searching the entire solution space for an optimal solution. Heuristic representsthe class of algorithms which makes more realistic assumptions about a priori knowl-edge concerning process and system loading characteristics [7, 8].

Generally, mapping tasks on distributed servers is an NP-hard problem, andworkflow scheduling as an optimization problem produces large scheduling over-head, especially for problems with two-dimensional constraints such as time andcost [9, 10]. The most well-known goal considered for workflow scheduling is mini-mizing the makespan of the application. Although many research papers have ad-dressed this problem [9, 11], in the economic scheduling, cost reduction along withsatisfying the deadline is very important and that should be taken into considera-tion in workflow scheduling [12]. Consequently, traditional approaches for schedulingtasks in grid community are no longer suitable for utility grids. Therefore, some newmethods have been proposed in the past years to fulfill this requirement [4, 6, 9,10, 13, 14, 15, 16, 17, 30]. Many recent approaches in workflow scheduling considercritical path as a hint to assign sub-deadlines to the tasks [11, 13, 17, 18, 19], butdeadline distribution with those methods is not efficient enough to decrease deadlineviolations. Another disadvantage of the previously presented scheduling methods is


the lack of priority between tasks in the mapping step. To overcome these short-comings, we develop a new scheduling algorithm with two steps. In the first step,an efficient method for initial distribution of tasks deadlines is presented, and in thesecond step, scheduling priority for the task with minimum slack time is consideredto reach a better result. In order to evaluate the proposed algorithm and compareit with others, we simulate three types of well-known workflows under various as-sumptions and system configurations. Simulation results show the advantage of theproposed algorithm in comparison with four most-cited recent algorithms.

The remainder of this paper is organized as follows. In Section 2, some relatedwork done on scheduling problem, especially on workflow scheduling in grid envi-ronments, is presented. In Section 3, the scheduling problem in general case and itsdetails in our context are described. The main proposed critical chain algorithm to-gether with other sub-methods is presented in Section 4. In Section 5, experimentalresults obtained from simulation are given. Finally, Section 6 concludes the paperand presents the future work which can be done in this research field.

2 RELATED WORK

There are several research works addressing the problem of mapping workflows onmultiprocessors [18, 20, 21]. However, some constraints like communication delaysand specifically budget issues on economic grids make the previously done researchwork on multiprocessor systems inefficient when they are applied to the grids.

Foster et al. [22] have described a General-purpose Architecture for Reservationand Allocation (GARA) that supports QoS specification. Dogan et al. [23] havestudied the scheduling of a set of independent tasks considering some QoS factorssuch as reliability, security and timeliness. Tabbaa et al. [24] have presented a newscheduling algorithm for DAG applications in clusters. The algorithm considersthe failure of resources and tries to schedule tasks to the cluster servers to toler-ate the faults occurred in the system. Entezari-Maleki et al. [25] have proposeda genetic-based task scheduling algorithm to minimize the makespan of grid ap-plications. The algorithm proposed in [25] only considers the makespan as a QoSfactor. However, there are few papers considering the cost of scheduling as a QoSfactor. Kardani-Moghaddam et al. [26] have proposed a hybrid genetic algorithmand variable neighborhood search which uses a fitness function to balance betweenmakespan and execution cost of each scheduling solution. Agrawal et al. [27] have ex-plored linear workflow scheduling for linear workflows, and found an approximationalgorithm to maximize throughput in the one-port model. Moreover, they provedthat finding a schedule respecting a given period and a given latency is NP-hard.

Yu et al. [9] have proposed the deadline-MDP algorithm that divides a DAG topartitions and then distributes the deadline over the partitions. Finally, deadline-MDP algorithm tries to locally optimize the cost for each partition using Markovmodels. It has been shown that deadline-MDP algorithm outperforms previousmethods such as DTL and greedy cost [9, 10]. The genetic algorithm was used to


optimize the time of scheduling under budget constraint in [6]. Zhao et al. [28] haveproposed two algorithms to schedule workflows with budget constraints. The firstalgorithm initially schedules all tasks to faster servers and then reschedules sometasks to meet the desires. Similarly, the second algorithm assigns each task to itscheapest server, and reschedules the tasks to the faster and more expensive serversas long as the budget is acceptable. According to Yuan et al. [29], Deadline BottomLevel (DBL) is a simple and efficient heuristics for workflow scheduling. In thismethod, all tasks are divided into several groups based on their bottom level witha backward method. The overall deadline is distributed over the groups consideringmaximum processing cycle of tasks in that group. All tasks in a group inherita unique deadline of the corresponding group. Unlike the DTL method [9], the starttime of each task is determined by the latest finish time of its immediate predecessorsinstead of the finish time of the last task in the previous level. Although DBL andDTL are effective and efficient, these algorithms show poor performance in firmdeadlines. Yuan et al. [10] have presented the Deadline Early Tree (DET) method.First, they create Early Tree which is a spanning tree for primary schedule. Thenthey exploit dynamic programming to assign time window to each critical task, andconsequently, find time window for non-critical tasks. Finally, the method tries toassign cheaper servers to each task according to its time window. The number ofservers was assumed to be unlimited which is unrealistic assumption in most cases.

Cost-effective scheduling of deadline-constrained applications have been also in-vestigated in hybrid clouds [15, 30, 31, 32, 33, 34]. Henzinger et al. [15] have designeda framework to handle cost-time trade-off in economic workflow scheduling calledFlexPRICE. They tried to present the cost-time curve to enable users to select theappropriate deadline with an acceptable price. In fact, FlexPRICE was presented tosolve cloud workflow programming, but the type of the problem is similar to the gridcomputing. Fard et al. [35] have introduced a pricing model and a truthful mech-anism for scheduling single tasks considering monetary cost and completion time.With respect to the social cost of the mechanism, they extended the mechanism fordynamic scheduling of scientific workflows. Calheiros et al. [31] have presented anarchitecture for coordinated dynamic provisioning and scheduling which is able tocost-effectively complete applications within their deadlines. They considered thewhole organization workload at individual tasks level, and their accounting mech-anism was used to determine the share of the cost of utilization of public cloudresources to be assigned to each user. Poola et al. [30] considered deadline and bud-get constraints as QoS demanded by users, and tried to design a robust algorithmfor scheduling of scientific workflows.

Abrishami et al. [13] have proposed a partial critical path scheduling based onproperties of critical path. In deadline assignment step, the deadline is distributedover tasks, and then the cost of each allocation is locally optimized to providethe best possible result in each allocation. For deadline distribution, the methoditeratively selects a sequence of tasks in the DAG and assigns the deadline to eachmember of that sequence. For this assignment, authors apply three different policieson the deadline distribution method: optimized policy, decrease cost policy, and fair


policy. The optimized policy iteratively tests all feasible assignments and selects thebest one. It is obvious that this approach is time consuming, and it is not feasible forlarge-scale problems. The decrease policy is based on a greedy method which tries toapproximate the optimized policy. In this policy, each task is assigned to the fastestserver, and it is tried to decrease the cost by rescheduling a task to a cheaper server.The fair policy is similar to the decrease policy except that starting from the firsttask towards the last task in path, it substitutes the assigned server with the nextslower server without exceeding sub-deadline. This procedure continues iterativelyuntil no substitution can be made. According to the results reported in [13], theproposed algorithms show high performance in absence of server limitation.

3 PROBLEM DEFINITION

Directed Acyclic Graph (DAG) is one of the most acceptable models to repre-sent workflow applications. Let G(T,E) denote a DAG representing an applicationwhere T is the task set T = {t1, t2, . . . , tn} in which n is the number of all tasksin the application. Moreover, edge set E represents the edges of the related DAGand shows the control or data dependencies between the tasks. The notation eijdenotes an edge from the vertex ti to tj, and means that the task correspondingto the vertex tj requires input data or command produced by execution of task ti.Suppose that all tasks are topologically numbered in which each arc demonstratesthe priority of i < j, means that execution of task tj only depends on the taskswith lower numbers. The tasks having no input (output) edges are named entry(exit) tasks. For simplicity and without loss of generality, we suppose that alwaysthere is only one entry task in the application. If an application has more than oneentry task, we can simply add a dummy task (a task that requires no processing)to it to produce our DAG of interest. Similarly, we can do the same for exit taskin the graph. The number attached to each node represents the processing cycle ofthe corresponding task in the form of Million Instructions (MI). Also, the numberattached to each arc eij shows the amount of data which should be sent from tito tj. Figure 1 shows an example of DAG representation. In the graph representedin Figure 1, a node with index of i shows task ti.

A service model in the utility grid computing consists of Grid Service Providers(GSPs) in which each of them provides some servers with specific QoS. The cost ofprocessing in each server is proportional to the speed of process which means thatif the scheduler allocates a faster server to execute a task, the user has to pay morecost [9]. Each server supports limited number of task types. We consider each ofGSPs as a grid node. Assume that there are m service providers represented by set Swhere S = {s1, s2, . . . , sm}. Hence, for each task, there are several candidate serverswhich can execute the task. Assume Tij is the processing time of task ti executedon server sj, and Cij is its corresponding processing cost. If sj is not capable ofprocessing ti, we consider both Tij and Cij to simply being infinity. It is assumedthat dummy tasks can be processed on any server.


0

1 2

3 4

5

100,000 MI Type: A

426,000 MI Type: B

468,000 MI Type: A

84,000 MI Type: A

536,000 MI Type: C

968,000 MI Type: C

768MB 256MB

1GB 512MB 512MB

512MB 512MB

Figure 1. DAG representation of a sample workflow

In this paper, two well-known QoS measures in grids, execution time and cost,are considered. Therefore, our objective in this paper is to assign an appropriateserver to the tasks to execute them with the goal of minimizing the overall executioncost while both tasks’ precedence and application deadline are taken into account.To achieve this, we can consider workflow scheduling as an optimization problemwith trade-off between time and cost [36]. Let δ denote a given deadline showingthe latest possible finish time of the application or exit task. Let sti and fi denotethe start and finish times of task ti, respectively. Therefore, the workflow schedulingproblem can be formulated as Equation (1).

min Σi∈TΣ1≤k≤mCikxik

S.t.

Σ1≤k≤mxik = 1, i ∈ T,

fi < δ, i ∈ T,

fi − sti = TiS(ti), i ∈ T,

sti > fj + Delay(tj, ti), j ∈ imm preds(i),

xik ∈ {0, 1}, 1 ≤ i ≤ n, 1 ≤ k ≤ m,

S(ti) = S(tj)⇒ (sti > fj) ∨ (fi < stj), i, j ∈ T

(1)


where

xij =

{1, ti is assigned to sj,

0, otherwise.

The constraint Σ1≤k≤mxik = 1 in Equation (1) checks to make sure that thereis a unique executor for each task. Similarly, condition fi < δ ensures meeting theoverall deadline. Moreover, the constraint fi − sti = TiS(ti) checks the feasibilityof task execution on the server in a given time slice, where S(ti) is the server as-signed to execute task ti, and TiS(ti) is execution time of task ti on S(ti). Eachtask would be executed on a resource only if its required data is transferred to theresource. This constraint is checked by sti > fj + Delay(tj, ti), where Delay(tj, ti) isdata transmission time on the link between S(ti) and S(tj) which is computed asEquation (2).

Delay(ti, tj) =eij

bandwidth(S(ti), S(tj))(2)

where bandwidth(S(ti), S(tj)) denotes the bandwidth of the link between serversexecuting tasks ti and tj.

In our model, it is considered that the number of servers is limited and someof the servers are busy in some cases, so they cannot be allocated to the tasks.The constraint S(ti) = S(tj)⇒ (sti > fj) ∨ (fi < stj) checks if the same server isallocated to execute both tasks ti and tj. If it is, the start time of one of thetasks (e.g., task tj) has to be greater than the finish time of the another one (e.g.,task ti). De et al. [36] showed that the time-cost optimization problem for DAGscheduling is a Discrete Time-Cost Trade-off Problem (DTCTP). DTCTP is anNP-hard problem, and the best-known solutions use dynamic programming, andbranch and bound method to solve the problem. Unfortunately, these solutions areextremely time-consuming when the number of tasks and/or servers gets larger.

4 THE PROPOSED ALGORITHM

In order to efficiently schedule the tasks on the servers, we need an initial estimationof execution times of the tasks on servers. This estimation could help us to identifycritical tasks of the application and schedule them on fast servers to meet the overalldeadline of the application. So, in the first step, we propose an algorithm to dividethe deadline on all tasks. After applying the algorithm of the first step, each taskwill have its own deadline in which the task should be processed before that deadline.The algorithm of the first step works based on the concept of slack time that is well-known in multiprocessor scheduling community. We use slack time in the deadlinedistribution algorithm to obtain the critical path not only for the whole graph, butalso for all its sub-graphs. This method helps us to have fair deadline distribution.In the second step, in order to meet deadline as much as possible, critical tasksshould be scheduled on faster servers. So, it is a good idea to have task priority inmapping step. Therefore, some models of priority are used in the mapping step and


then dynamic programming technique is exploited to have an efficient sequentialtask scheduling.

After describing the totality of the proposed algorithm, details of the algorithmare provided in the following sections. Some notations used in the algorithm areintroduced in Section 4.1. Deadline distribution over all the tasks is describedin Section 4.2. In Section 4.3, an illustrative example of the proposed deadlinedistribution algorithm is provided. After deadline distribution phase, partitioningtechnique is explained in Section 4.4. This technique is used to have better mappingfor pipeline branches. Finally, in Section 4.5, the mapping algorithm based onpriority method is explained.

4.1 Basic Definitions

Minimum Execution Time (MET) for task ti refers to the minimum time for anexecution of task ti on the fastest available server which is capable of processingtask ti. Equation (3) shows MET calculation.

MET(ti) = mins∈S

ET (ti, s) (3)

where S and ET (ti, s) denote the set of servers in the system and the execution timeof task ti on server s, respectively. In fact, mins∈S ET (ti, s) is the execution time oftask ti on the fastest available resource for executing ti.

Minimum Transfer Time (MTT) for arc eij denoted by MTT(eij) refers to theminimum time for data transmission of eij on the maximum available bandwidth ofthe grid. The calculations of MTT(eij) is shown in Equation (4).

MTT(eij) = min∀S(ti),S(tj)∈S

Delay(ti, tj). (4)

Earliest Start Time (EST) for each task refers to the earliest possible time thatthe task can start its execution. In other words, EST shows earliest possible starttime of task ti if all predecessors of ti are executed on the fastest server(s). Similarly,Latest Finish Time (LFT) of a task refers to the latest possible finish time of thetask while executing all successors of that task on the fastest server does not causeabsolute deadline violation. Therefore, if execution of task ti terminates at the timeLFT + ε and then all other tasks are executed on the fastest server, the executionof entire application will be finished on the time δ + ε. We can compute EST andLFT as Equation (5) and Equation (6), respectively.

EST(tentry) = 0,

EST(ti) = maxtp∈imm preds(ti)

(EST(tp) + MET(tp) + MTT(epi)

), (5)

LFT(texit) = δ,

LFT(ti) = mintc∈imm succs(ti)

(LFT(tc)−MET(tc)−MTT(eic)

). (6)


Assume all predecessors and successors of task ti are planned to be executedon the fastest server(s), so we have a wide range of time to schedule ti. In otherwords, we are free to schedule task ti in each part of this time period. Subtractingminimum execution time of task ti from this time period, slack time of ti shown byST(ti), is obtained. In fact, ST(ti) is the maximum possible extra time for executingtask ti minus its minimum execution time. Equation (7) shows the slack time oftask ti.

ST(ti) = LFT(ti)− EST(ti)−MET(ti). (7)

4.2 Deadline Distribution

Assigning a sub-deadline to each task according to the overall deadline of an appli-cation is the main objective of this step. In other words, we wish to assign a timewindow to each task for execution. This time window is determined by ScheduleStart Time (SST) and Schedule Finish Time (SFT). During the deadline assign-ment, we are dealing with two types of tasks: assigned and unassigned tasks. Ifwe assign a sub-deadline to a task (specifying both the start and finish times forthe task), it is flagged as an assigned task, otherwise, it is called unassigned task.Dedicating the time interval of SST and SFT to the tasks does not mean that thetasks should be executed in these intervals; but these intervals give us an offlineapproximation of execution time of tasks based on other tasks within the workflow.The final goal of deadline distribution phase is assigning fair SST and SFT to eachtask. For this purpose, first, we have to compute earliest start time (EST) and latestfinish time (LFT) of each task. Note that, for the assigned tasks, we do not com-pute EST and LFT measures, because they are equal to SST and SFT, respectively.Deadline assignment procedure is represented in Algorithm 1.

This algorithm begins with computing MET and MTT for all tasks, and then it-eratively updates EST, LFT, and ST for each task and chooses a chain of unassignedtasks having minimum slack time. Considering this procedure, we certainly havea sequence of consecutive unassigned tasks that have minimum slack time. The firstelement of this sequence has assigned parent tasks and the last element has assignedchild tasks. We call this sequence as critical chain. In other words, a critical chainis a sequence of unassigned tasks in which each task ti is the immediate successor oftask ti−1 in the chain. Moreover, all tasks in the critical chain have the same slacktime which is the minimum one amongst slack times of all unassigned tasks withinthe application. If there is more than one critical chain with the aforementionedcharacteristic, one of them is arbitrarily chosen.

After selecting a critical chain, a time window can be assigned to each taskof the chain. Let CST (Chain Start Time) denote a maximum LFT of immediatepredecessors of the first task in the chain, and similarly, CFT (Chain Finish Time)denote a LFT of the last task in the chain. Now, we have a chain interval timeCFT–CST which should be distributed over chain’s tasks. We have already dis-tributed this time interval to all tasks of the chain considering MET of each task.Actually, SST of the first task in the chain is its CST. Therefore, to obtain SFT


Algorithm 1: Deadline Distribution

1 Input: Application

/* initialization */

forall ti ∈ Application doCompute MET(ti);Compute MTT(ti);

end

while (there are unassigned tasks) doUpdate EST and LFT for all unassigned tasks;minSlack←−∞;chain← null;

/* Check for critical sequence of task to schedule */

for i = 0 to i = numberOfTasks− 1 doif IsAssigned(ti) then

continue;endST(ti)←− LFT(ti)− EST(ti)−MET(ti) ;if ST(ti) < minSlack then

sequence←− null ;add ti to the end of sequence ;minSlack←− ST(ti);

else if TF (ti) = minSlack thenif sequence.lastItem is imm pred of ti then

add ti to the end of sequenceend

end

end/* distribute deadline over some partitions */

CST←− maxt∈imm Preds(Sequence.firstItem)

LFT(t);

CFT←− LFT(sequence.lastItem) ;

Distribute deadline (CFT− CST) over all partitions in sequenceproportional to MET(ti) ;

end

(sub-deadline) of a task, it is sufficient to only add time interval assigned to the taskto its SST. SST of the next task in the chain is also equal to SFT of the previoustask in that chain. It turns out that SFT of the last task in the chain is equal toCFT. We mark all the tasks in the chain as assigned tasks and update unassignedtasks’ EST and LFT measures and then continue with selecting a new critical chain.


This procedure goes on until all tasks are assigned. It should be mentioned thatfor each assigned task, EST and LFT are considered as SST and SFT, respectively.By assigning time window to the last task, sub-deadline assignment procedure iscompleted.

Finding computational complexity of the proposed deadline distribution algo-rithm is beyond the scope of this paper, but intuitively, it can be seen that in theworst-case, the algorithm assigns sub deadline to at least one task in each iteration.Moreover, the complexity of each iteration for updating SST and SFT is O(n2),where n is the number of tasks in the workflow. Therefore, the computational com-plexity of the algorithm is O(n3), but the upper bound can be more tight thanstated. This bound is pessimistic and the practical complexity for real applicationsis much less than this value. From another point of view, we can consider that thecomplexity of updating SST, SFT and ST is O(e), where e is the number of depen-dency relations between the tasks, or the number of edges in DAG representing ofthe workflow. Therefore, the computational complexity of the algorithm is O(ne).Since e can take (n)(n + 1)/2 in worst case, the computational complexity of ourproposed algorithm is O(n3).

4.3 An Illustrative Example

Figure 2 is an example of critical chain deadline distribution for workflow applicationshown in Figure 1 while a sample grid environment with the specification presentedin Table 1 is considered. Each processing node in the sample grid has its ownspecific processing power indicating million instructions that the node can processper second (MIPS). Obviously, different processing nodes have different usage pricesbased on their processing speeds. In this sample, bandwidths of all links amongthe servers are assumed to be 512 Mbps. In Figure 2 A), MET and MTT measuresare shown for each task. As an example, task t2 has 426 000 MI with computationrequirement type of B. The fastest server which can execute this task is server s2.Therefore, MET(t2) is 426 000 MI/2 000 MIPS = 213 S. Similarly, we can computeMTT and MET measures for all tasks.

We can compute sub-deadlines for this example in three iterations shown inFigure 2 B) to Figure 2 D). The measures EST, LST and ST are computed for alltasks, and critical chain is obtained (e.g., {t0, t2, t4, t5}). This step is shown inFigure 2 B). In the first iteration, chain interval is equal to whole deadline (1250).Since, task t2 needs 225 seconds to be completed (MET + MTT = 213 s + 12 s =225 s), and task t4 needs 125 seconds, the time assigned to task t2 is 225/125 = 9/5times greater than the time assigned to task t4. Similarly, the time interval will bedistributed over all members of the critical chain. Iteratively, we update EST, LSTand ST measures to obtain another critical chain (chain {t1} in Figure 2 C)). Since,there exists only one task in critical chain shown in Figure 2 C), the whole timeinterval is assigned to task t1, and SST and SFT measures of task t1 are set to 50and 500, respectively (Figure 2 D)). Finally, SST and SFT measures related to taskt3 are forced to be 500 and 750, respectively.


0

1 2

3 4

5

MET = 25s

MET=213s

MET = 117s

MET= 21s

MET=134s

MET = 242s

MTT=12s MTT=4s

MTT=16s MTT=8s MTT=8s

MTT=8s

MTT=8s

A 0

1 2

3 4

5

EST = 0s LFT = 650s FT = 625s

EST = 37s LFT = 875s FT = 625s

EST = 258s LFT = 1000s FT =625s

EST = 383s LFT = 1250s FT = 625s

EST = 29s LFT = 850s FT = 800s

EST = 66s LFT = 1000s FT = 800s

Current Critical Chain

Other Dependencies

B

0

1 2

3 4

5

SST = 0s SFT = 50s

EST = 54s LFT = 500s FT = 425s

EST = 91s LFT = 750s FT = 525s


Other Dependencies

SST = 50s SFT = 500s

SST = 500s SFT = 750s

SST = 750s SFT = 1250s

C



0

1 2

3 4

5

SST = 0s SFT = 50s

EST = 91s LFT = 750s FT = 525s

Other Dependencies


SST = 500s SFT = 750s

SST = 750s SFT = 1250s

D

Figure 2. Deadline distribution over a sample workflow

4.4 Partitioning Technique

Partitioning is a method to optimally solve branches with several sequential tasksin grid workflows [9]. In partitioning model, tasks are divided into two differentcategories: simple tasks and synchronization tasks. If a task has at most one im-mediate predecessor, and at most one immediate successor, then the task is namedsimple task, otherwise, it is called synchronization task. We use the word branch

Server Server TypeProcessing Power Price per Second

(MIPS) ($)

s1 A, B, D 1 000 0.001

s2 B, C, E 2 000 0.004

s3 A, C, D, E 4 000 0.016

Table 1. Sample grid specification with three processing nodes


to refer to several sequential simple tasks (a sub-graph like a pipeline workflow).The mapping problem of branches can be solved using dynamic programming andhidden Markov model. In Markov model, each of the states can be considered asa triple like [termination time, number of scheduled tasks in a branch, last scheduledserver ]. Afterwards, one can calculate value of each state with the help of previousstates using dynamic programming. In this problem, the value of each Markov stateis the cost of scheduling. The detailed description of partitioning technique can befound in [9]. Moreover, it should be mentioned that partitioning does not take partin deadline distribution phase of the proposed algorithm. This technique is usedbesides mapping phase. First, we partition the workflow in the branches, and then,in the mapping phase, we allocate a resource to all tasks in the branch with dynamicprogramming if we encounter a branch.

It remains a subtle point that dynamic programming is very time consuming,and calculation time extremely increases with increasing the problem size. Size ofthis problem depends on three different factors: the number of branch tasks, numberof servers, and termination time range. The number of branch tasks and servers areconstant values and if those numbers get higher, dynamic programming becomesinefficient. In almost all cases in our problem, we encounter few task brancheswith limited number of servers which can process each task. Therefore, dynamicprogramming can appropriately tolerate those factors in our case. As mentionedearlier, the third factor is termination time. If the time interval is wide, then we cansegment the time to larger time pieces. Actually, we can handle larger problems inexchange for a bit of accuracy using segmentation.

4.5 Mapping Based on Priority

In the mapping phase, we try to map servers to the tasks to optimize the overallcost besides meeting the overall deadline. To do this, it is tried to have a localoptimization in server allocation process to hopefully reach the global optimization.In the mapping phase, we iteratively pass over three steps. Mapping algorithmfirstly selects a ready task (partition), a task that all of its immediate predecessorshave been executed. Secondly, it updates the start time of unscheduled tasks tothe minimum finish time of immediate predecessor plus delay of incoming link ofpredecessor. When there are one or more processors for processing a task beforeexpiring its deadline, we choose the cheapest processor among those. If there is noserver available to fulfill task’s deadline, a server with minimum deadline violationis selected. Since both the number and power of the servers existing in the systemare limited, it is important to have a priority algorithm to select a ready task to beassigned to the limited servers. The following three different methods for priority-based mapping are used in this paper:

Simple priority: A ready task with lower task ID has higher priority for schedulingin this method. Using this mechanism, we can simulate random priority.


Start time priority: A ready task with minimum start time has higher priorityduring the mapping phase. The main idea of start time priority method is thatif a task has a lower start time, then there are probably more tasks waitingfor that task, so it should be scheduled as soon as possible. The other ideabehind start time priority is that, as soon as there is a ready task available forscheduling, it may be better to schedule it and do not wait for other tasks tobecome ready.

Slack time priority: For each ready task, a measure named slack time is com-puted using SST and SFT measures. Equation (8) computes slack time for eachtask.

ST(ti) = SFT(ti)− SST(ti)−MET(ti). (8)

In slack time priority, the ready task with minimum slack time is prior to bescheduled. The main idea of slack time priority method is that if a task haslower slack time, then scheduling this task is more likely to be critical, and post-poning its scheduling may lead to local sub-deadline violation and, consequently,possible global deadline violation.

According to the results presented in Section 5, slack time priority shows betterperformance in comparison with two other approaches. Finally, for the sake ofbrevity, we only present the performance analysis of the slack time priority.

5 PERFORMANCE EVALUATION

Accurate performance evaluation not only depends on the perfect implementationof the proposed and benchmark algorithms, but also it highly relies on the testdata and experimental setup. In this section, it is described how workflow test setsare generated, and what are the main experimental setups. After that, the resultsobtained from the experiments are presented.

5.1 Generating Workflows

Three types of common workflow structures, pipeline, parallel and hybrid work-flows [9, 10, 11, 13, 14, 19, 37], are considered in this paper. These structures areshown in Figure 3. Pipeline workflow consists of numbers of tasks in a sequentialorder (Figure 3 A)). Parallel workflows include multiple pipelines with some middlesynchronizer tasks (Figure 3 B)). Hybrid workflows are combinations of pipeline andparallel workflows in an arbitrary order (Figure 3 C)). Structure of pipeline workflowsis simple, but many factors influence on construction of parallel and hybrid work-flows. Most important factors in parallel workflow construction are the maximumwidth of graph (the number of parallel pipeline chains), and maximum number ofpossible sub-pipeline tasks. We set the maximum width and the maximum pipelinelength to 10 and 20, respectively. There is no constraint on choosing other numbersfor width and maximum pipeline length. We just choose those numbers according


A) Pipeline B) Parallel C) Hybrid

Entry Task Simple Task Exit Task

Figure 3. Different types of random workflow structures

to previously done similar work [10]. Generating hybrid workflow structures is morecomplex. There are some papers addressing this problem [37, 38, 39]. They studyparametric features of each generation method such as average critical path lengthand average edge. In [39], a tool for random graph generation, named TGFF, whichis approximately fair in graph generation was presented. According to study doneby Cordeiro et al. [37], the Max-in/Max-out degree method is one of the best meth-ods for workflow generation. Hence, TGFF is used in this paper to produce hybridworkflows with Max-in/Max-out method. For TGFF method, the Max-in (Max-out)parameters are considered from 1 to 3 (Maxin(Maxout) ∈ {1, 2, 3}), and the numberof tasks varies from 30 to 1000. Figure 3 C) shows a sample output of TGFF withspecified parameters. In addition to random workflows, we compare the proposedmethod with other algorithms applying the standard workflows used in [13] calledCyberShake, Montage, LIGO, Gnome, and SIPHT, which are real workflows in thescientific or business communities.

5.2 Experimental Setup

We consider details of real environments in our experiments such as grid structure,network bandwidth, processing power of each processing node, largeness of work-flows, deadline determination and so forth. All assumptions made in this work areconsistent with previously done research work [6, 9, 10, 13, 14]. In this section, wehave a glance on implementation details.

Workflow Specification: The structures of workflows are chosen randomly as il-lustrated in Section 5.1. From the viewpoint of granularity, we consider three


types of workflows: small, medium and large. The number of tasks vary from30 to 200, from 200 to 600, and from 600 to 1 000 in small, medium and largeworkflows, respectively. For the heterogeneity of workflows, it is assumed thateach task has specific processing requirements. We assume there are 15 differenttypes of tasks. Moreover, the size of each task varies from 100 000 MI to 900 000MI in this study. For each workflow, there are several exit tasks and only oneentry task. Each task can be executed as soon as its corresponding resourcereceives the required data for processing that task. The input/output data forthe task changes from 10 MB to 1 GB, as well.

Grid Specification: To simulate the server heterogeneity in grid network, we con-sider 15 different types of services, each service supported by 10 different gridservers. Network bandwidth between each two grid servers is considered to bein the range of 200 Mbps to 512 Mbps. Grid server processing power shown byMIPS (Million Instructions Per Second) varies from 1 000 MIPS to 5 000 MIPSfor each node. For each processing node there is a price proportional to its powerranging from 0.001 $ to 0.025 $. This means that executing a task on a serverwith a processor n times faster than another, imposes n times more cost to thescheduling.

Hardware Specification: We run the simulation on a regular PC with Intel R©CoreTM i5-4200M (3M Cache, up to 3.10 GHz) and 4 GB RAM. Based on thishardware configuration, each test case runs in 1 to 2 seconds (except for hugepipeline workflows), and consequently, each test set containing more than 500different test cases runs in less than 30 minutes. For a pipeline workflow testcase with 200 tasks, it takes up to 5 minutes to run. It turns out that the timerequired to run the simulation reduces by using more powerful hardware.

Deadline Assignment: Each workflow is tested with 9 different deadlines, i.e.,δn = Tmin×θ, where Tmin is the minimum completion time for that workflow onthe specified grid, and θ ∈ {1, 1.05, 1.1, 1.15, 1.2, 1.5, 2, 2.5, 3}. Practically, thereis no need to consider θ > 3, because in that range, all tasks are delivered tolower and cheaper processing nodes, and approximately all methods show thesame efficiency. Determining Tmin is an NP-hard problem, so we use HEFTgreedy algorithm [28] to estimate minimum completion time.

5.3 Result Analysis

We compare our method with four recent most-cited methods: PCP Fair, PCPDecrease, DTL, and DBL [9, 13, 29]. As mentioned earlier, we use three typesof workflows to do experiments. Each of random workflows is tested by over 800instances to achieve more reliable results. Each instance is tested by 5 differentmethods with 8 different deadlines, and the experiment is done with near 100 000iterations. Three different priority-based mapping methods are tested, but for thesake of brevity, only one of the results is reported here.


DBL DTL Decrease PCP Fair PCP Critical Chain

Small 0 0 0 0 100PipeLine Medium 0 0 0 0 100

Large 0 0 0 0 100

Small 10.4 10.5 0.33 0.22 89Parallel Medium 11.88 11.78 2.78 1.56 83.69

Large 17 17 0.22 13.44 69.33

Small 26.02 25.4 0.34 11.23 48.87Hybrid Medium 20.09 22.10 2.91 22.27 31.58

Large 29.89 24.66 0 16.31 36.62

Table 2. Best result percentage

In our study, an algorithm is superior to another if it appropriately satisfies thefollowing three properties:

Best results: if results of two different schedulings violate the deadline, the resultthat has the minimum time is the best result for our case study. If none ofthem has deadline violation, the result with the minimum scheduling price isthe best one. An algorithm with maximum percentage of best results is namedbest suitable algorithm in our study.

Deadline Violation Rate (DVR): if an algorithm has minimum deadline viola-tion rate, then it is superior to the others.

Average cost: the algorithm with minimum average cost is preferred to the others.

The best result percentages of all algorithms for all test sets are given in Ta-ble 2. As it can be seen in Table 2, from the aspect of the best results percent-age, the proposed critical chain method is dominant. This means that for most ofthe applications, critical chain shows the best performance compared to the otherbenchmarks.

In pipeline workflows, critical chain uses dynamic programming for branches,and computes optimal solution; therefore, this method is dominant in this typeof workflows. Since the algorithm produces best result in pipeline workflows, thepercentage of the test cases on which any other method can produce best result iszero. This is shown in Table 2 by elements 0 inside cells related to the pipelineworkflows. It should be mentioned that since the scheduling problem consideredin this paper belongs to the set of NP-hard problems, and there is no polynomialsolution to solve it, we tried to solve it by dynamic programming that is very timeconsuming. Therefore, the best solution with 100% confidence in pipeline workflowsis achieved by imposing extra scheduling time on the scheduler. Dynamic program-ming imposes heavy computational overhead to the scheduler, so that, computingthe best solution for over 200 sequential tasks in a reasonable time is impractical forschedulers. For the parallel workflows, our algorithm is dominant too with a slightdecrease in performance in comparison with other algorithms. Parallel workloads


contain many pipeline chains and our algorithm uses dynamic programming andfair deadline distribution to receive this performance. If we consider a very longlength for a parallel workload, then we have to solve many long pipeline problemsto schedule the long-length parallel workload, and as a result, we may encounter theperformance problem described above. It is worthwhile to mention that schedul-ing parallel workloads has no polynomial optimal solution. Therefore, performancedegradation in comparison with pipeline workflows is reasonable. In the hybridworkflows, the pipeline chains are in minority, so dynamic programming cannot beuseful further, but our algorithm still shows relatively good performance. We be-lieve that our deadline distribution method has an important role to achieve this.Based on the simulations done, we found that increasing the size of the graph doesnot considerably change the results. Hence, the proposed algorithm shows the sameperformance for different workflow sizes. In other words, the structure of a workflowhas much greater impact on the results. We continue to analyze the algorithms inthe view of deadline violation rate and average cost properties for each of the work-flow structures. For the sake of brevity, we depict only two classes of diagrams foreach workflow type. The left-side diagrams in Figure 4 show average cost resulted byalgorithms. Since these diagrams show the average cost for all small, medium, andlarge applications, they should be normalized to match with each other in a singlediagram. The Normalized Cost (NC) is computed as Equation (9).

NC =Scheduling Cost

Minimum Scheduling Cost. (9)

After scheduling each workflow, the cost resulting from the scheduling is dividedby the minimum scheduling cost obtained from a greedy algorithm and then NC isachieved. The minimum scheduling cost is obtained from greedy scheduler thatassigns each task to the cheapest resource with unlimited deadline. The right-side diagrams in Figure 4 show the deadline violation rate of the algorithms. Ifa violation rate of one algorithm is α with β as a deadline factor, it means that α isthe probability of deadline violation by applying that scheduling algorithm withβ × Tmin as a deadline. All algorithms try to schedule all tasks to faster serverswhenever deadlines are firm (deadline = 1). This tends to increase the schedulingcost resulted from all methods. Moreover, the rate of deadline violation increases inthis situation. Expanding the deadline associated with each task, and as a result,increasing the overall deadline of the workflow, deadline violation rate and executioncost resulted from all algorithms decrease.

For the pipeline (Figure 4 A) and Figure 4 B)) and parallel (Figure 4 C) andFigure 4 D)) workflows, DVR and NC for critical chain get minimum values com-pared to the other algorithms. This shows that the best method for schedulingthese types of workflows amongst all of the algorithms implemented in this paperis critical chain. For hybrid workflows (Figure 4 E) and Figure 4 F)) DVR in alldeadlines and NC for soft deadlines are minima in critical chain method, but inthe firm deadlines, NC for critical chain is slightly more than DBL algorithm. It is


A- NC for Pipeline Workflows

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C- NC for Parallel Workflows

D- DVR for Parallel Workflows

E- NC for Hybrid Workflows F- DVR for Hybrid Workflows

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Figure 4. Comparison of Normalized Cost (NC) and Deadline Violation Rate (DVR) re-sulted from all algorithms for random workflows

predictable, because in a firm deadline, critical chain tries to not exceed deadlineby scheduling tasks on the faster (consequently more expensive) resources. Finally,it can be stated that DVR and average NC for critical chain are minima in compar-ison with other algorithms. Hence, the proposed critical chain method outperformsprevious methods from the viewpoint of best results, deadline violation rate, andaverage cost.

As mentioned in Section 5.1, we apply our proposed algorithm on realistic stan-dard workflows presented in [13]. The results obtained from applying the proposedalgorithm on medium size workflows including CyberShake, Montage, and LIGO arepresented in Figure 5, and similarly the results gained from workflows with largeand small sizes, Gnome and SIPHT, are presented in Figure 6. As can be seen in


B – DVR for CyberShake Workflow

A – NC for CyberShake Workflow

D – DVR for Montage Workflow

C – NC for Montage Workflow

F – DVR for LIGO Workflow

E – NC for LIGO Workflow

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Figure 5. Comparison of Normalized Cost (NC) and Deadline Violation Rate (DVR) re-sulted from all algorithms for standard benchmark workflows with medium sizes

both Figure 5 and Figure 6, our algorithm dominates other algorithms in averageby minimizing DVR and reducing the scheduling cost.

6 CONCLUSIONS AND FUTURE WORK

The basic principle used in utility computing and grid computing is identical which isproviding computational resources as a service. One of the most important problemsin such environments is scheduling deadline constrained bag-of-tasks applications oncomputational resources. In this paper, we study the deadline of applications as one


B – DVR for Gnome Workflow (Large size)

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C – NC for SIPHT Workflow (Small size)

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Figure 6. Comparison of Normalized Cost (NC) and Deadline Violation Rate (DVR) re-sulted from all algorithms for standard benchmark workflows with large and small sizes

of the more interesting factors in grid computing, and try to deliver a service withspecified QoS and minimum cost. Therefore, our problem can be considered asa time-cost trade-off problem. To solve this optimization problem, two-step criticalchain heuristic is presented. In deadline distribution phase, the algorithm appliesa fairer mechanism to better deadline distribution, which finally, leads to lowercost of service. In the second step named resource allocation phase, resources areallocated to the tasks efficiently according to the priority of tasks. Finally, applyingthe proposed method to different scenarios and system settings, it is shown that theproposed approach outperforms other similar existing methods.

Critical chain is applicable in other distributed computing domains such asclouds. In cloud systems, customers can select their desired service based on theirbudget and time constrains, and pay for using these services. Therefore, the prob-lem of scheduling workflow applications on limited resources considering time andbudget constraints is an interesting problem in clouds. So, one possible extensionto this work is applying the proposed critical chain method on scheduling workflowsin cloud infrastructures considering their specific characteristics and requirements.

Critical path-based deadline distribution has deficiency in bursting tasks. Inother words, if the workflow width becomes unpredictable during the specified timeperiod, the deadline distribution based on critical path with any available methodwould be inefficient. So, one can modify the proposed method to handle this type


of workflows. Furthermore, considering the map-reduce structure as a new type ofworkflows and modifying the proposed method to have high performance in map-reduce workflows can be assumed as another objective. Evaluating other perfor-mance measures such as resource utilization and taking server setup cost into ac-count can be considered as an important direction for improvements of the currentwork.

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Alireza Dehlaghi-Ghadim is currently Ph.D. candidate in theDepartment of Electrical and Computer Engineering, Universityof Tehran, Tehran, Iran. He received his M.Sc. degree fromSharif University of Technology, Tehran, Iran, in 2012, and hisB.Sc. degree from Iran University of Science and Technology,Tehran, Iran in 2009. His main research interests are grid andcloud computing, task scheduling algorithms, load balancing andresource allocation methods.

Reza Entezari-Maleki is Post-Doctoral Researcher in theSchool of Computer Science at Institute for Research in Funda-mental Sciences (IPM) in Tehran, Iran. He received his Ph.D. incomputer engineering (software discipline) from Sharif Univer-sity of Technology, Tehran, Iran in 2014, and M.Sc. and B.Sc.degrees in computer engineering (software discipline) from IranUniversity of Science and Technology, Tehran, Iran in 2009 and2007, respectively. He visited the Seoul National University inSeoul, South Korea, Duke University in NC, USA, and Insti-tuto Superior Tecnico in Lisbon, Portugal in 2012, 2013, and

2015, respectively, His main research interests are performance/dependability modelingand evaluation, grid and cloud computing, and task scheduling algorithms.

AliMovaghar is Professor in the Department of Computer En-gineering at Sharif University of Technology in Tehran, Iran andhas been on the Sharif faculty since 1993. He received his B.Sc.degree in electrical engineering from the University of Tehranin 1977, and M.Sc. and Ph.D. degrees in computer, informa-tion, and control engineering from the University of Michigan,Ann Arbor, in 1979 and 1985, respectively. He visited the Insti-tut National de Recherche en Informatique et en Automatiquein Paris, France and the Department of Electrical Engineeringand Computer Science at the University of California, Irvine in

1984 and 2011, respectively, worked at AT & T Information Systems in Naperville, IL in1985–1986, and he was teaching at the University of Michigan, Ann Arbor in 1987–1989.His research interests include performance/dependability modeling and formal verificationof wireless networks and distributed real-time systems. He is a senior member of the IEEEand the ACM.

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